In one type of process for the production of stress rated lumber, the bending modulus of elasticity (MOE) of each wood board is measured. MOE is an intrinsic material property, which, together with board dimensions, determines bending stiffness. MOE and other characteristics of the board are used to categorize it into a grade. Off-line quality control tests help to assure that boards in the higher grades have the properties required in those grades. Examples in North America include MSR (Machine Stress Rated) and MEL (Machine Evaluated Lumber) lumber grades.
Measurements of bending MOE in a board require a bending span of some length over which to apply and/or measure bending forces and deflections. The resulting determination of bending MOE, denoted Em, is a composite result, which represents the intrinsic MOE values at points along the board corresponding to the length segment of the board coinciding with the bending span. Because each measurement is a composite, the values Em so obtained are smoothed versions of the underlying pointwise MOE values. Measurement of MOE by bending is inherently a smoothing operation.
In wood boards, this smoothing is an issue because of widely varying structural characteristics within a board. Knots and grain angle deviations are among the characteristics affecting structural value. Structural failures of wood boards are often associated with local characteristics of the boards rather than with their average properties. Smoothing, as a result of bending measurements, can mask variations in local, i.e. pointwise, values that, if estimated more accurately, could identify structural problems. Enhanced estimation accuracy will be useful in the determination of further manufacturing, processing or use of the board, including as input to other processes for the same purposes.
A production-line machine used in North America and elsewhere for measuring MOE of wood boards is known as the HCLT (High Capacity Lumber Tester) and is manufactured by Metriguard Inc in Pullman Wash. The HCLT follows specification of U.S. Pat. No. 5,503,024 (Bechtel et al. 1996).
In the HCLT, a wood board to be tested enters and passes longitudinally through it, typically at speeds exceeding 6 m/s (1200 ft/min). Overlapping length segments for measurement points along the board are each presented sequentially to two bending sections and bent downward in a first bending section and then upward in a second bending section. Bending spans in each bending section are defined by rollers at support points that apply bending deflections and forces to the board. At the center of each bending section, the force at a roller support is measured by an electronic load cell. Measured forces from the two bending sections are averaged, a differential time delay being required to match measurements from the same length segment along the board. By averaging downward and upward forces, the measurements are compensated for lack of straightness in the board. Each MOE measurement is determined by scaling an average force measurement.
In the HCLT, detail of MOE along the board is lost due to the inherent smoothing previously mentioned.
The following is a description of related prior art that attempts to alleviate the smoothing effect of bending measurements and obtain greater detail.
Prior Art 1. Shorter Bending Spans
One obvious method for estimating MOE more locally than in the HCLT, and hence for reducing smoothing, is to reduce lengths of bending spans. The central part of a fill seven-support bending span in the HCLT covers a board segment length of 1219 mm (48 inch), although the total extent between first and last supports is 2032 mm (80 inch). For this discussion of the possibility of using shorter spans, weighting of effects from board segments outside the 1219 mm central part of the span may be considered negligible.
Other production-line machines have been used with bending spans having length 900 mm (35 in). However, these machines have only three support points as opposed to the seven supports of the HCLT. The multiple supports of the HCLT approximate fixed end conditions. Comparison has shown that a seven-support span of the HCLT provides a more localized result than shorter (900 mm) spans using center-loaded simple supports (Bechtel and Allen 1995).
It would be possible to redesign and reduce the bending span lengths used in any of the bending machines. However, precision of roller positioning, pitch buildup on rollers, bearing precision, compression of wood fibers at roller-to-wood contact points, machine rigidity, and wood surfacing tolerances all become more critical with shorter spans. Reduced accuracy of bending measurement is a consequence of reduced bending span lengths. Present designs have considered these tradeoffs, and present-day equipment for testing boards having dimension in the direction of bending in the usual range of about 35 to 45 mm (including 1.5 inch) are a reasonable compromise. Regardless of bending span length chosen, bending measurements are still inherently composites and hence smoothed versions of local values.
Prior Art 2. (Bechtel 1985)
This prior art research showed that the smoothing is described mathematically as applying to the reciprocal of MOE called compliance. It was shown that the measured compliance along a beam is the convolution integral of the unknown local compliance function and a weighting function h(x) hereinafter called “span function”. Span function h(x) was derived for the special case of a three-support, center-loaded bending span having length L. The span function h(x) shows how much each local compliance value is weighted into the measured compliance. Thus, h(x) determines the weights in a weighted average of the local compliances in their contribution to the measurement. For the simple three-support case mentioned, the span function is given by:       h    ⁡          (      x      )        =      {                                                                      (                                  12                                      L                    3                                                  )                            ⁢                                                (                                                            L                      2                                        -                                                                x                                                                              )                                2                                      ,                                                                            x                                      ≤                          L              /              2                                                                        0            ,                                                otherwise            .                              From Prior Art 2, with some change in independent variable definition, measured compliance Cm(w) may be written as the convolution integral (using * to indicate convolution):                                           C            m                    ⁡                      (            w            )                          =                              h            ⁡                          (              w              )                                *                      C            ⁡                          (              w              )                                                              =                              ∫                                          -                L                            /              2                                                      +                L                            /              2                                ⁢                                    h              ⁡                              (                w                )                                      ⁢                          C              ⁡                              (                                  w                  -                  x                                )                                      ⁢                          ⅆ              x                                          where w is a point of measurement on the beam, measured from the beam's leading end and identifying where the bending span is being applied.
Prior Art 2 suggests recovering the local compliance values C(w) by use of the well-known complex convolution theorem, which turns convolution into multiplication if the Fourier Transform is taken. Specifically:{overscore (C)}m(f)={overscore (h)}(f){overscore (C)}(f) where the overbar indicates Fourier Transform, and f is a frequency variable. The idea is to divide and take the inverse Fourier Transform (ift) to obtain the local compliance estimates C*(w) as:                                           C            *                    ⁡                      (            w            )                          =                  ift          ⁡                      (                                          C                _                            ⁡                              (                f                )                                      )                                                  =                  ift          ⁡                      (                                                                                C                    _                                    m                                ⁡                                  (                  f                  )                                                                              h                  _                                ⁡                                  (                  f                  )                                                      )                              where the superscript asterisk * indicates estimate. Carrying out the indicated operations on real data gives estimates that are dominated by high frequency noise. The reason for this may be seen if measurement noise v(w) at w is additive to the measurement of compliance according to the model;Cm(w)=h(w)·C(w)+v(w) Then, the above result for C*(w) becomes:                                           C            *                    ⁡                      (            w            )                          =                  ift          ⁡                      (                                                                                C                    _                                    m                                ⁡                                  (                  f                  )                                                                              h                  _                                ⁡                                  (                  f                  )                                                      )                                                  =                  ift          ⁡                      (                                                                                                      C                      _                                        ⁡                                          (                      f                      )                                                        ⁢                                                            h                      _                                        ⁡                                          (                      f                      )                                                                      +                                                      v                    _                                    ⁡                                      (                    f                    )                                                                                                h                  _                                ⁡                                  (                  f                  )                                                      )                                                  =                              C            ⁡                          (              w              )                                +                      ift            ⁡                          (                                                                    v                    _                                    ⁡                                      (                    f                    )                                                                                        h                    _                                    ⁡                                      (                    f                    )                                                              )                                          At high frequencies, the Fourier Transform {overscore (v)}(f) of the noise is significant and the Fourier Transform {overscore (h)}(f) of the span function is very small. The second term involving noise can be very large.Prior Art 3. (Foschi 1987)
Using the deconvolution suggestion of Prior Art 2, Foschi reduced noise by truncating high frequency noise components. Using simulated data, Foschi concluded that noise problems could be alleviated and the deconvolution process made practical.
Prior Art 4. (Richburg et al. 1991)
Richburg, et al. reported that, while Foschi's method gave close approximations to simulated data, attempts to verify his method with experimental data were unsuccessful.
Prior Art 5. (Lam et al. 1993)
Lam et al. elaborated on noise reduction and refined Foschi's frequency truncation method.
Prior Art 6. (Pope and Matthews 1995)
Pope and Matthews further expanded on Foschi's frequency truncation approach. They correlated bending strength with estimates of local MOE nearest the failure and also with machine measured MOE nearest the failure. Pope and Matthews concluded that the estimated local MOE results from any of Foschi's, Lam et als.' or their own work gave no meaningful improvement in correlation over the direct machine measurement of MOE.
Prior Art 7. (Bechtel and Allen 1995)
A figure was presented illustrating span functions for both a seven-support bending span of an HCLT and for a three-support bending span. Derivation of the span function for the seven-support system was not disclosed, nor was it correct for the figure presented (a correct general approach for determining span functions is a part of the present specification for estimating local compliance). Because of the scale and resolution of the figure in this prior art disclosure, errors caused by the incorrect span function derivation are not easily apparent. The seven-support HCLT span function was compared with span functions for three-support, center-loaded bending spans. The conclusion that measurement resolution of an HCLT seven-support bending span is better than the resolution of shorter three-support, center-loaded bending spans was correct.
Further Discussion of Prior Art
The span function for a three-support, center-loaded bending span was mistakenly used to analyze the CLT (Bechtel 1985) which is an earlier machine similar to the HCLT. The difference in span functions for the CLT and the HCLT compared to the three-support span function is significant. Bechtel reported the error, apparently in time to prevent erroneous application to real data from the CLT. Others (Foschi 1987, Richburg et al. 1991, Lam et al. 1993, Pope and Matthews 1995) have processed signals using only span functions for three-support, center-loaded spans. The prior art includes no data analyzed with span functions for the more complicated seven-support bending spans. No optimal method exists in the prior art for estimating local compliance (or MOE). In one of the prior art papers, the span function for a three-support, center loaded bending span may have been used to weight local MOE values directly rather than their reciprocals. There is no theoretical support for this use of the span function.
All of the above referenced prior art that attempt local estimation by deconvolution reduce noise by truncation of information above a cutoff frequency. While that method is helpful, it is ad hoc and suboptimal.
Local compliance estimation by deconvolution in Prior Art 2, 3, 4, 5, and 6 requires use of Fourier Transforms. In practice, this leads to complications, particularly for points near the ends of a beam. Fourier methods cause artifacts at discontinuities, e.g. Gibb's Phenomenon (Guillemin 1949), that must be considered and compensated or otherwise accounted for.
Although a summary of prior art research (Pope and Matthews 1995) is pessimistic about usefulness of local MOE estimation, the above discussion identifies reasons why the prior art efforts were not particularly successful.
Preliminaries
The new method of this specification uses equations, called beam equations, developed from flexural loading theory found in elementary texts on mechanics of materials, e.g. (Higdon et al. 1960). As opposed to the usual analyses, MOE is not assumed constant along the beam. Local MOE, namely E, or E(w) to indicate its dependence on position, appears in the denominators of the beam equations along with cross-sectional area moment of inertia I as a product EI. In some cases I is also a function of position w along the beam, and then it is written I(w). Details for computation of I from cross-sectional geometry may be found in basic references. In the case where the beam has a rectangular cross-section, I=γ1γ23/12, where γ1 and γ2 respectively are orthogonal cross-sectional dimensions perpendicular to and in the plane of bending.
Because E appears with I as a product in the beam equations, this product may be treated as a single parameter that can vary with position along the beam. Thus, the new method may be applied to beams with variable E, I, or EI product along the length.
Conventional analyses are built around MOE. MOE is an intrinsic material property defined as the ratio of stress to strain at a point. The reciprocal ratio of strain to stress could equally well have been defined and used. In this specification, it is asserted that the span function weighting discussed above, should be applied to the reciprocal of E. In cases where just I varies with position, the weighting is applied to the reciprocal of I. In cases where E and I both vary and hence are considered together as a product, the weighting applies to the reciprocal of the EI product. Use of reciprocals of the normally considered quantities is a natural outcome of the development and occurs because E and I are in the denominators of the beam equations. Modulus of elasticity appears in the discussion only because it is so deeply embedded by convention in the minds and literature of those working in the field.
To ease the complexity of notation and maintain generality of the result, the notation C is used to indicate any of the reciprocals 1/E, 1/I, or 1(EI) depending on whether E, I, or EI are considered as varying with position w. The new method is equally applicable for any of these cases. Similarly, measured compliance Cm can be any of 1/Em, 1/Im, or 1/(EmIm). In the case where one or the other of E or I is constant, that constant value can be removed from the integrals involved and combined with other constants. Unless otherwise indicated, in the development of the equations, where it is helpful to check dimensions, it is assumed that C=1/(EI), and Cm=1(EmIm) Symbols Cm and Em without subscript refer to local compliance and MOE values, and symbols Cm and Em with subscript “m” refer to measured compliance and MOE values. When written as functions of an argument, as in C(w) or Cm(w), the argument is taken as the distance from the leading end of a beam, which, in this specification, is the right end with w measured leftward. All compliance and MOE measurements in this specification are assumed to be in bending. The use of the abbreviation MOE is reserved for modulus of elasticity in general.
For a bending span, measured compliance Cm(w) is a weighted composite of local compliance values C(u) for u in a neighborhood of w. The weighting with which each C(u) contributes to the measurement Cm(w) is defined by a span function specific to the bending span. Knowledge of the span function is essential in the optimal estimation of local compliance values.
The phrase “bending span” refers to a beam support configuration, with supports at specified points in the beam's longitudinal direction. At the supports, forces and deflections to the beam are applied and or measured in a direction substantially perpendicular to the beam's length and substantially in the plane of bending. Moments and slopes are not specifically considered except at beam ends, because they can be applied and or measured by forces and deflections at closely spaced pairs of support points.
Applied to a beam at a measurement point w on the beam, bending span implies a derived functional relationship expressing the measured compliance Cm(w) as a functional of local compliance along the beam. Corresponding with the bending span for each measurement point w is a length segment of the beam, usually the distance between first and last of the bending span support points, from which local compliance values affect Cm(W). Thus, Cm(W) is associated with a particular bending span, measurement point, length segment, and span function, as well as the local compliance values.
For different measurement points w along the beam, the bending span, and hence span function, can change. For example, in an HCLT bending section there are seven supports. When a measurement point w is near the center of a wood board, all supports are active; but, if w is 30 inch from an end of the board, only five supports are active. The bending spans for the two cases are different and different span functions are applicable. The new method uses corresponding sequences of measured compliance values Cm and span functions to optimally estimate local compliance values C.
High-speed present day measuring equipment for testing wood boards uses force measurements to provide results in units of MOE. In that case, the taking of reciprocals will provide a sequence of compliance measurements Cm. Values of compliance obtained either by measuring them directly or by taking reciprocals of measured MOE are referred to as measured compliance values or just compliance measurements and designated with the symbol Cm or Cm(w) as a function of measurement position.
In the result, local MOE estimates E*(w) may be obtained from local compliance estimates C*(w) by taking reciprocals, e.g. E*(w)=1/C*(w), and modified with a correction to be described. The superscript asterisk is used to indicate estimates.
In the case of estimated values, the reciprocal relationship between E*(w) and C*(w) must be considered with caution. Each estimated compliance value C* is taken as the mean value of its estimator C, and quality of the estimate may be taken as either the variance of the estimator or its coefficient of variation COV. Coefficient of variation of an estimator is defined as the ratio of standard deviation divided by the mean. Estimator E for local MOE is obtained from the reciprocal distribution of C, i.e. from the distribution of E=1/C. But, the mean of a reciprocal distribution is not equal to the reciprocal of the mean, although it may be close if COV is small. It can be shown from statistical methods (Papoulis 1991) that a first correction for the mean and COVE of estimator E may be given approximately in terms of the mean and COVC of estimator C according to:             E      *        ≅                  1                  C          *                    ⁢              (                  1          +                      COV            C            2                          )                        COV      E        ≅                  COV        C            ⁡              (                  1                      1            +                          COV              C              2                                      )            If COVC<<1, then E*≅1/C*, and COVE≅COVC. Accuracy is improved if the correction factor (1+COVC2) is used as shown. In this specification, optimal compliance estimates C* and COVC are the result. When E estimates E* and COVE are required, the above relationships with correction are used.
The preferred and alternative embodiments are examples where the method applies to apparatus used for bending measurements of wood boards. However, it can be applied equally well to bending measurements of any elongated beam. For example, it could be applied to bending measurements on panel products by apparatus (Lau and Yelf 1987, Dunne and Lau 2000) which bend panels. The method does not require a continuous movement or rolling of a beam relative to a bending span, although that is the means often employed. The method is applicable to any type of beam and not just those made of lignocellulosic material.
Local compliance at w is representative of the cross-section at w, and is a composite of intrinsic compliance values for all points within that cross-section. How each of the compliance values within the cross-section contributes to local compliance at w may be found elsewhere; e.g. (Bechtel 1985). No attempt is made in this specification to use bending compliance measurements to deduce the intrinsic compliance at points within a cross-section. Rather, they are used to estimate local compliance values C at points along the length of a beam.
Measurements are samples of a continuous function Cm(w) at a sequence of measurement points, i.e. at discrete values of the argument w, separated by a sampling increment d. Typically, by nature of bending tests, the sequence of measurement points begins and ends at a distance greater than some multiple of d from beam ends.
Estimates of local compliance by the new method are identified with discrete cross sections along the beam length. However, each may be taken as representative of local compliance at points within half the sampling increment d of a cross section. Thus, the estimates represent the compliance for elements of a subdivision of the beam length.
In the above description and for the preferred embodiment description, the subdivision is regular-, that is, each sampling increment has length d. This restriction to a regular subdivision, as opposed to an irregular subdivision wherein sampling increments are not all the same length, simplifies data processing and is a convenient but not a necessary restriction of the method.
Objects and Advantages of the New Method
The new method is able to provide optimal estimates for general bending span configurations, not just for simple cases where span function is easily obtainable by prior art methods. As a necessary part of the method, a general procedure is disclosed for computing span functions.
The new method optimally estimates local compliance values C along a beam and provides a measure of estimation quality for each estimate.
The new method provides estimates of local compliance out to the ends of a measured beam, thereby addressing one of the objections of bending measurements. Estimation quality is reduced at points near the beam ends, as one would expect, because fewer compliance measurements are used in the estimation at those points. Also, the amount of the contribution to any measured compliance from a local compliance value near the ends of a tested beam is small. But, for the first time, a measure of estimation quality is provided and a priori information is used to help extend estimation to these previously unavailable segments of the beam.
The new method can accommodate changes in the bending span as a wood board moves through a machine such as the HCLT. When the leading end of the board enters a bending section of an HCLT, useful data are obtained before all seven supports are engaged. Data are obtained as the bending span number of supports is first 5, then 6, and then 7 corresponding to the leading end of the board engaging additional supports. Similarly, the number of supports changes from 7 to 6 to 5 as the trailing end progresses through the bending section and disengages from supports. The changing number of supports and hence bending span is addressed herein by using bending spans and hence span functions specific for each measurement point. There is no known approach to account for this non-stationarity of bending span with any of the Fourier methods.
The new method works in the same domain as the data and does not require transformations to the frequency domain and back. It thereby avoids these processing steps and the necessity of dealing with discontinuity effects at board ends.
The new method is recursive. Each estimate is used in obtaining the next. The optimal compliance estimate for a point of estimation along a board may be computed as soon as the compliance at that point no longer contributes to any remaining compliance measurements. In an HCLT, except for a small delay due to a decimation preprocessing step to be described, this is as soon as the element being estimated exits the last data producing bending span defined in the second bending section. This helps to make the process practical in real time on a production line. The prior art Fourier Transform methods require that the entire data stream be processed together.
The new method allows inclusion of available a priori information in practical models of the random process describing beam compliance measurements. Parameters describing an autoregressive random process for local compliance of the beam may be learned from autocorrelation values obtained from compliance measurements for sufficiently long lags. Bending span parameters (weighting coefficients) for inclusion in the moving average part of an autoregressive moving average (ARMA) model of beam compliance measurements are determined by the disclosed procedure for computing span function and computing weights from span function. Statistics of measurement noise may be included as part of the model, and other confounding noise sources may be modeled.
Inclusion of available a priori information is important in obtaining the best estimates possible. For example, vibration noise sometimes contributing to the output signal from the HCLT machine may be included in the model, and the estimated component of compliance measurement attributed to vibration noise may be excluded from local compliance estimates. Moreover, the estimated vibration noise component may be developed into a measure of machine performance and as an indicator for maintenance. The model framework allows other noise processes to be included as they become known with consequent improvement in compliance estimation quality.
The new method can use additional measurements that are affected by the local compliance values to improve quality of local compliance estimation. While the embodiments describe a scalar compliance measurement sequence, the new method allows vector measurements. Other scalar measurements, that are affected by local compliance, may be used as additional components of a measurement vector. For example, it may prove desirable to uncouple the prior art process of averaging measurements from a second section HCLT bending span with delayed measurements from a first section HCLT bending span. Then, the separate measurements from the two bending sections would be included in a more complex model having a two-dimensional vector measurement sequence.
Measurements that are affected by beam parameters other than local compliance may be included and these other parameters estimated by the same method framework. This can lead to a more comprehensive indicator of beam quality than provided by compliance measurements alone.
The new method provides a measure of estimation quality. Variances of the compliance estimators are part of the computed results. Either variance or COV may be used as a quality measure.